MATHEMATICS PROJECT R311 something on godf
- 2. f Linear Programming Problem Name: Ruhaan Mari Class: 12-A(Science) Roll No: 19 CC: 246285
- 3. f Certificate This is to inform that Ruhaan Mari, a student of class XII A Science, of the Bishop’s School, Camp Pune has successfully completed the project Work on the topic Using any suitable data, find the Optimum cost in the manufacturing problem By formulating a linear programming problem (LLP) Under the guidance of Mrs. Seema Negi In partial fulfilment of the ISC Mathematics, 2025-2026 conducted by CISCE, New Delhi Signature of External Examiner Signature of Internal Examiner
- 4. f Acknowledgement I would like to express my gratefulness to our principal Mr. McPherson and Mrs. Seema Negi of the Mathematics Department for grating me this opportunity to undertake this project. Their guidance and support have been invaluable. I also extend my heartfelt appreciation to my classmates whose encouragement and contribution towards this project have been significantly insightful and helpful. Finally, I wish to express my profound thanks to my family for their unwavering support, encouragement and understanding which has assisted me through the challenges and demands of this academic endeavour.
- 5. f INDEX Sr No. TOPIC 1 Introduction 2 Types of LLP 3 AIM and Application 4 Question 1 5 Question 2 6 Question 3 7 Question 4 8 Question 5 9 Limit of LLP 10 Conclusion
- 6. f INTRODUCTION Linear Programming (LP), also known as a Linear Programming Problem (LPP), is an optimization technique for finding the best outcome (such as maximum profit or minimum cost) of a linear objective function, subject to linear equality or inequality constraints. In other words, an LPP seeks the optimal (highest or lowest) value of a linear objective function given a set of linear relationships among the variables. It is widely used in fields like mathematics, economics, and engineering to optimize resource allocation and planning under constraints. Components of an LPP: An LPP typically consists of decision variables (the unknowns to solve for), an objective function (a linear function of the decision variables to be maximized or minimized), and a set of constraints (linear equations or inequalities the variables must satisfy). For example, an LPP might have two variables x and y, maximize z = c₁x + c₂y, subject to constraints like a₁x + b₁y ≤ c₁, etc. All decision variables are usually assumed non-negative. The feasible region of the LPP is the set of all variable values satisfying the constraints; this region is always a convex polygon or polyhedron defined by the intersecting half-spaces of the linear inequalities
- 7. f Types of LPP Manufacturing problems: AIM: Maximize profit or minimize production cost while considering resource limitations (like labor, machines, materials). Variables: Let P₁, P₂, … = Number of units to produce of Product 1, Product 2, etc. Example: A company makes two products: Chairs and Tables. Each chair needs 2 hours of labor and 3 units of wood. Each table needs 3 hours of labor and 2 units of wood. The company has 100 hours of labor and 90 units of wood. Profit is ₹30 per chair, ₹20 per table. → Formulate an LPP to maximize profit.
- 8. f Types of LPP Diet Problems: AIM: Minimize the cost of food while meeting nutritional requirements (like calories, protein, vitamins, etc.). Variables: Let F₁, F₂, … = Quantity (in grams or servings) of Food Item 1, Food Item 2, etc. Example: A diet must provide at least 500 calories and 50g protein per day. Rice provides 250 cal and 10g protein per serving; costs ₹10. Milk provides 150 cal and 20g protein per glass; costs ₹15. → Find the cheapest food combination to meet the nutritional goal.
- 9. f Types of LPP Transportation Problem: AIM: Minimize total shipping or delivery cost from multiple origins (factories) to multiple destinations (warehouses/customers). Variables: Let Sᵢⱼ = Number of units to ship from Source i to Destination j Example: A company has two warehouses (A & B) and three stores (X, Y, Z). Each warehouse has a supply limit; each store has a demand requirement. The shipping cost between each pair (e.g., A to X, A to Y, etc.) is known. → Formulate an LPP to minimize shipping cost.
- 10. f Types of LPP Assignment Problems: AIM: Assign tasks, jobs, or people in such a way that the total cost or time is minimized, or efficiency is maximized. Variables: Let Aᵢⱼ = 1 if Worker i is assigned to Job j, else 0 Example: There are 3 workers and 3 tasks. Each worker takes a different time to complete each task. Assign each worker to exactly one task such that total time is minimized. → Formulate a 0-1 LPP (binary assignment problem). insta
- 11. f AIM and APPLICATION of LLP The aim of solving an LPP is to find the optimal (maximum or minimum) value of the objective function that satisfies all constraints. . In practical terms, this means allocating limited resources (materials, time, money, etc.) in the best possible way according to the chosen criterion (e.g. maximize profit, minimize cost or time). Applications of LPP span numerous industries and activities: transportation and logistics planning, production and manufacturing scheduling, communications network design, energy resource allocation, diet and nutrition planning, and more. Many operations-research problems reduce to LPP: for example, optimizing crew scheduling in airlines, planning delivery routes, balancing chemical production, or even balancing a diet – all can be formulated and solved as LPPs Aim (Objective): Find the best possible (maximal or minimal) value of a linear objective function subject to the given constraints Applications: Widely used in transportation (logistics), energy and telecommunications (network flows), manufacturing (production planning), and more. LPP helps in planning, routing, scheduling, assignment, and design problems, providing efficient decision support in business, economics and engineering
- 12. f QUESTION 1
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- 14. f QUESTION 3
- 15. f QUESTION 4
- 16. f QUESTION 5
- 17. f. Limitation of Linear Programming Linearity Assumption: All relationships (objective and constraints) must be linear. In reality, many processes are nonlinear (e.g. economies of scale, diminishing returns). If linearity is violated, LPP may give misleading results. Certainty (Fixed Coefficients): LPP assumes all coefficients (costs, resource availabilities, etc.) are known exactly and do not change. In practice, these values can be uncertain or variable. If the true parameters change, the LPP solution may no longer be valid. Fractional Solutions (Divisibility): LPP allows fractional values of decision variables. If the real decision must be integer (e.g. people, cars), the LPP optimum may not be practical and may need rounding. Rounding can break optimality or feasibility. Single Objective: LPP handles only one objective function. Real problems often have multiple, possibly conflicting goals (e.g. maximize profit and maximize market share). Standard LPP cannot optimize multiple objectives simultaneously. Scale and Complexity: Very large LPPs (with many variables/constraints) can be complex and computationally intensive. Also, once formulated, LPP models are not very flexible: small changes require re-solving the entire model
- 18. f Conclusion Linear Programming (LPP) is a fundamental optimization tool for solving allocation problems with linear relationships. It provides a systematic way to maximize or minimize an objective subject to resource constraints. We have covered its definition, typical problem types, goals and applications, solution methods (graphical and simplex), and common solution cases (unique, multiple, infeasible, unbounded). We also discussed concepts like duality and noted key limitations such as the need for linear, certain data and single objectives. Overall, LPP’s blend of mathematical rigor and practical utility makes it a powerful method for decision-making in business, engineering, and economics.
- 19. f Bibliography Authoritative texts and resources on linear programming en.wikipedia.org byjus.com geeksforgeeks. orggeeksforgeeks.